Preparation

Cut breasts from duck and discard skin and all visible fat. Reserve legs and carcass for another use.

Brown duck neck (halved if necessary) in a dry 2- to 3-quart heavy saucepan over moderate heat, turning once, about 10 minutes total. Transfer to a plate. Pat duck breasts dry and season with salt and pepper, then add to pan and brown over moderate heat, turning once, about 8 minutes total. Transfer to plate with duck neck.

Add Armagnac to pan and deglaze by boiling, scraping up any brown bits, until most of liquid is evaporated. Add 2 cups water, shallots, garlic, sliced carrot, thyme, bay leaf, kosher salt, peppercorns, and duck breasts and neck with any juices on plate and cook at a bare simmer, covered, 1 hour. Discard neck, then transfer breasts to a plate and cool. Shred duck meat, then chop.